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\begin{document}

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\section*{M5-brane}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory}

[[!include string theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{as_a_greenschwarz_type_sigmamodel}{As a Green-Schwarz type sigma-model}\dotfill \pageref*{as_a_greenschwarz_type_sigmamodel} \linebreak
\noindent\hyperlink{as_a_black_brane}{As a black $p$-brane}\dotfill \pageref*{as_a_black_brane} \linebreak
\noindent\hyperlink{AtAnOrbifoldSingularity}{At an orbifold singularity}\dotfill \pageref*{AtAnOrbifoldSingularity} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{worldvolume_theory}{Worldvolume theory}\dotfill \pageref*{worldvolume_theory} \linebreak
\noindent\hyperlink{branes_inside_the_m5}{Branes inside the M5}\dotfill \pageref*{branes_inside_the_m5} \linebreak
\noindent\hyperlink{dimensional_reduction}{Dimensional reduction}\dotfill \pageref*{dimensional_reduction} \linebreak
\noindent\hyperlink{holographic_dual}{Holographic dual}\dotfill \pageref*{holographic_dual} \linebreak
\noindent\hyperlink{7dCSDual}{Conformal blocks and 7d Chern-Simons dual}\dotfill \pageref*{7dCSDual} \linebreak
\noindent\hyperlink{RestrictionOfTheCField}{Restriction of the supergravity $C$-field}\dotfill \pageref*{RestrictionOfTheCField} \linebreak
\noindent\hyperlink{m5brane_charge}{M5-brane charge}\dotfill \pageref*{m5brane_charge} \linebreak
\noindent\hyperlink{AnomalyCancellation}{Anomaly cancellation}\dotfill \pageref*{AnomalyCancellation} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak
\noindent\hyperlink{survey}{Survey}\dotfill \pageref*{survey} \linebreak
\noindent\hyperlink{ReferencesAsBlackBrane}{Black brane description}\dotfill \pageref*{ReferencesAsBlackBrane} \linebreak
\noindent\hyperlink{unwrapped}{Unwrapped}\dotfill \pageref*{unwrapped} \linebreak
\noindent\hyperlink{wrapped_on_hyperblic_3manifolds}{Wrapped on hyperblic 3-manifolds}\dotfill \pageref*{wrapped_on_hyperblic_3manifolds} \linebreak
\noindent\hyperlink{ReferencesSigmaModelDescription}{$\sigma$-Model description}\dotfill \pageref*{ReferencesSigmaModelDescription} \linebreak
\noindent\hyperlink{ReferencesWorldvolumeTheory}{Worldvolume theory}\dotfill \pageref*{ReferencesWorldvolumeTheory} \linebreak
\noindent\hyperlink{ReferencesHopfWessZuminoTerm}{Hopf-Wess-Zumino term}\dotfill \pageref*{ReferencesHopfWessZuminoTerm} \linebreak
\noindent\hyperlink{ReferencesAnomalyCancellation}{Anomaly cancellation}\dotfill \pageref*{ReferencesAnomalyCancellation} \linebreak
\noindent\hyperlink{double_dimensional_reduction_to_d4brane}{Double dimensional reduction to D4-brane}\dotfill \pageref*{double_dimensional_reduction_to_d4brane} \linebreak
\noindent\hyperlink{open_m5branes}{Open M5-branes}\dotfill \pageref*{open_m5branes} \linebreak
\noindent\hyperlink{nonabelian_2form_fields}{Nonabelian 2-form fields}\dotfill \pageref*{nonabelian_2form_fields} \linebreak
\noindent\hyperlink{more_on_the_holographic_description}{More on the holographic description}\dotfill \pageref*{more_on_the_holographic_description} \linebreak
\noindent\hyperlink{more_on_the_algebraic_topology}{More on the algebraic topology}\dotfill \pageref*{more_on_the_algebraic_topology} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In [[11-dimensional supergravity]] the [[brane]] electrically charged under the [[supergravity C-field]] is the [[M2-brane]]/[[membrane]]. The dual under [[electric-magnetic duality]] is the M5-brane.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{as_a_greenschwarz_type_sigmamodel}{}\subsubsection*{{As a Green-Schwarz type sigma-model}}\label{as_a_greenschwarz_type_sigmamodel}

As a [[Green-Schwarz sigma-model]]: \hyperlink{BLNPST97}{BLNPST 97}

\hypertarget{as_a_black_brane}{}\subsubsection*{{As a black $p$-brane}}\label{as_a_black_brane}

As a [[black brane]] solution of [[11-dimensional supergravity]] the M5-brane is given (\hyperlink{Gueven92}{Gueven 92}) by the [[spacetime]] $\mathbb{R}^{5,1} \times (\mathbb{R}^5-\{0\})$ with [[pseudo-Riemannian metric]] given by

\begin{displaymath}
g = H^{-1/3} g_{\mathbb{R}^{5,1}} \oplus H^{2/3} g_{\mathbb{R}^5-\{0\}}
\end{displaymath}
for $H = 1 + \frac{1}{r}$ and $r$ the distance in $\mathbb{R}^5$ from the origin, and with [[field strength]] of the [[supergravity C-field]] being

\begin{displaymath}
F = \star_{\mathbb{R}^5} \mathbf{d}H
  \,.
\end{displaymath}
This is a $1/2$-[[BPS state]] of 11-dimensional supergravity.

The [[near horizon geometry]] of this spacetime is [[anti de Sitter spacetime|AdS7]]$\times$[[4-sphere|S4]]. For more on this see at \emph{[[AdS-CFT]]}.

[[!include black branes in supergravity -- table]]

\hypertarget{AtAnOrbifoldSingularity}{}\subsubsection*{{At an orbifold singularity}}\label{AtAnOrbifoldSingularity}

 More generally for 1/2 BPS black M5-branes, the near horizon geometry is $AdS_7 \times S^4/G$, where $G$ is a [[finite subgroup of SU(2)]] ([[ADE classification|ADE subgroup]]) acting by left multiplication on the [[quaternions]] $\mathbb{H}$ in the canonical way, under the identitfication $S^4 \simeq S(\mathbb{R}^5) \simeq S(\mathbb{R}\oplus \mathbb{H})$ (\hyperlink{MFF12}{MFF 12, section 8.3}).

While this geometric discussion in \hyperlink{MFF12}{MFF 12, section 8.3} works for all the [[finite subgroups of SU(2)]], folklore has it that in [[M-theory]] the M5-branes appear only at A-type singularities, while the more general [[6d (2,0)-superconformal field theories]] for all possible [[ADE-singularities]] appear only after passage to [[F-theory]] (\hyperlink{ZHTV14}{ZHTV 14, p. 3}).

On the other hand, when placing the M5 at an MO5-[[orientifold]] (\hyperlink{Witten95}{Witten 95}) its worldvolume theory breaks from $(2,0)$ to $(1,0)$-supersymmetry and all ADE-singularities should be allowed.

(\ldots{})

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{worldvolume_theory}{}\subsubsection*{{Worldvolume theory}}\label{worldvolume_theory}

the [[worldvolume]] theory of the M5-brane is the [[6d (2,0)-superconformal QFT]].

This [[worldvolume]] theory involves [[self-dual higher gauge theory]] of the [[nonabelian cohomology|nonabelian]] kind (\hyperlink{Witten07}{Witten07}, \hyperlink{Witten09}{Witten09}): the fields are supposed to be [[connections on a 2-bundle]]($\sim$ [[gerbe]]), presumably with structure [[2-group]] the [[automorphism 2-group]] $AUT(G)$ of some [[Lie group]] $G$.

For instance in the proposal of (\hyperlink{SSW11}{SSW11}) one sees in equation (2.1) \emph{almost} the data of an $\mathfrak{aut}(\mathfrak{g})$-[[2-groupoid of Lie 2-algebra valued forms|Lie 2-algebra valued forms]].

\hypertarget{branes_inside_the_m5}{}\subsubsection*{{Branes inside the M5}}\label{branes_inside_the_m5}

The M5-brane admits two solitonic excitations ($p$-branes within branes)

\begin{itemize}%
\item $p = 1$: the [[self-dual string]]


\item $p = 3$: the [[3-brane in 6d]] (see there for more)



\end{itemize}
See also

\begin{itemize}%
\item [[M2-M5 brane bound state]]

\end{itemize}
\hypertarget{dimensional_reduction}{}\subsubsection*{{Dimensional reduction}}\label{dimensional_reduction}

On [[Kaluza-Klein mechanism|dimensional reduction]] of [[11-dimensional supergravity]] on a circle the M5-brane turns into the [[NS5-brane]] and the [[D-brane|D4-brane]] of [[type II string theory]].

The [[Kaluza-Klein mechanism|compactification]] of the 5-brane on a [[Riemann surface]] yields as [[worldvolume]] [[theory (physics)|theory]] [[N=2 D=4 super Yang-Mills theory]]. See at \emph{\href{N%3D2+D%3D4+super+Yang-Mills+theory#ConstructionByCompactificationOf5Branes}{N=2 D=4 SYM -- Construction by compactification of 5-branes}}.

\hypertarget{holographic_dual}{}\subsubsection*{{Holographic dual}}\label{holographic_dual}

The [[AdS/CFT correspondence]] for the 5-brane is $AdS_7/CFT_6$ and relates the [[6d (2,0)-superconformal QFT]] to [[7-dimensional supergravity]] obtained by [[Kaluza-Klein mechanism|reduction]] of [[11-dimensional supergravity]] on a 4-[[sphere]] to an asymptotically 7d [[anti de Sitter spacetime]].

\hypertarget{7dCSDual}{}\subsubsection*{{Conformal blocks and 7d Chern-Simons dual}}\label{7dCSDual}

The [[self-dual higher gauge theory|self-dual 2-connection]]-field (see there for more details) on the 6-dimensional [[worldvolume]] M5-brane is supposed to have a [[holographic principle|holographic]] description in terms of a [[higher dimensional Chern-Simons theory|7-dimensional Chern-Simons theory]] (\hyperlink{Witten96}{Witten 1996}). We discuss the relevant ``fractional'' [[quadratic form]] on [[ordinary differential cohomology]] that defines the correct [[action functional]].

Let $\hat G$ be the [[circle n-bundle with connection|circle 3-bundle with connection]] on a 7-dimensional manifold $X$ with boundary the M5-brane, thought of as the [[Kaluza-Klein mechanism|compactification]] of the [[supergravity C-field]] from [[11-dimensional supergravity]] down to [[7-dimensional supergravity]].

As discussed there, the [[higher dimensional Chern-Simons theory|7-dimensional Chern-Simons theory]] [[action functional]] on these 3-connections is

\begin{displaymath}
\hat G_4 \mapsto
  \exp(i \int_X \hat G_4 \cup \hat G_4)
  \,,
\end{displaymath}
where

\begin{itemize}%
\item $\exp(i \int_X (-))$ is the [[higher holonomy]] / [[fiber integration in ordinary differential cohomology]] from $X$ to the point


\item of the [[Beilinson-Deligne cup product]] [[circle n-bundle with connection|7-connection]] $\hat G_4 \cup \hat G_4$.



\end{itemize}
The space of [[states]] of this 7d theory on the M5 worldvolume $\partial X$ would be the space of [[conformal blocks]] of the [[6d (2,0)-supersymmetric QFT]] on the [[worldvolume]].

Except, that it turns out that the [[first Chern class]] of the corresponding [[prequantum line bundle]] is \emph{twice} that required from [[geometric quantization]].

Therefore the above action functional is not yet the correct one, but only a fractional version of it is. However, the class $G_4 \cup G_4$ in [[integral cohomology]] has in general no reason to be divisible by 2.

This is related to the fact that as a [[quadratic form]] on the [[ordinary differential cohomology]] group $\hat H^4(X)$, the above is not a [[quadratic refinement]] of

\begin{displaymath}
(\hat G, \hat G') \mapsto \exp(i \int_X \hat G \cup \hat G') 
  \,,
\end{displaymath}
but of twice that. In (\hyperlink{Witten96}{Witten 1996}) it was argued, and later clarified in (\hyperlink{HopkinsSinger}{Hopkins-Singer}), that instead the action functional should be replaced by a proper [[quadratic refinement]].

This is accomplished by shifting the center of the [[quadratic form]] by a lift $\lambda \in H^4(X, \mathbb{Z})$ of the degree-4 [[Wu class]] $\nu_4 \in H^4(X, \mathbb{Z}/2)$ from 0 to $\frac{1}{2}\lambda$.

(For that to make sense in [[integral cohomology]], either the [[Wu class]] $\lambda$ happens to be divisible by 2 on $X$, or else one has to regard it itself as a twisted differential character of sorts, as explained in (\hyperlink{HopkinsSinger}{Hopkins-Singer}). For the moment we will assume that $X$ is such that $\lambda$ is divisbible by 2.)

Since $X$, being a [[spacetime]] for [[supergravity]], admits (and is thought to be equipped with) a [[spin structure]], by the discussion at [[Wu class]] it follows that $\lambda$ is the [[first fractional Pontryagin class]] $\frac{1}{2}p_1$

\begin{displaymath}
(\frac{1}{2}p_1 \; mod \; 2) 
   \; 
     =
   \; 
   \nu_4 \in H^4(X, \mathbb{Z}/2)
  \,.
\end{displaymath}
By the very definition of [[Wu class]], it follows that for any $\hat \alpha \in \hat H^4(X)$ the combination

\begin{displaymath}
\hat \alpha \cup \hat \alpha + \hat \alpha \cup \hat \lambda = Sq^4(\hat \alpha) - \hat \alpha \cup \hat \lambda
  \; =\;  0 \;  mod  \; 2
\end{displaymath}
is divisible by 2.

Therefore define then the modified quadratic form

\begin{displaymath}
\exp(i S^\lambda) 
   \; : \; 
  \hat a \mapsto \exp i \int_X 
  \frac{1}{2}
  \left(
    \hat a \cup \hat a
    + 
    \hat a \cup \hat \mathbf{\lambda}
  \right)
\end{displaymath}
(see [[differential string structure]] for the definition of the differential refinement $\hat \mathbf{\lambda} = \frac{1}{2}\hat \mathbf{p}_1$), where, note, we have included a global factor of 2, which is now possible due to the inclusion of the integral lift of the Wu class.

Notice that where the [[equations of motion]] of the original [[action functional]] are $\hat a = 0$, those of this shifted one are $\hat a = - \frac{1}{2}\hat \mathbf{\lambda}$. One may therefor calls $-\frac{1}{2}\lambda$ here a \emph{background [[charge]]} for the 7-d Chern-Simons theory.

This is now indeed a [[quadratic refinement]] of the [[intersection pairing]]:

\begin{displaymath}
\exp i \left( 
    S^\lambda\left(\hat a + \hat b \right)
    - 
    S^\lambda\left( \hat a \right)
    - 
    S^\lambda\left( \hat b \right)
    + 
    S^\lambda\left( 0 \right)
  \right)
  = 
  \exp i \int_X ( \hat a \cup \hat b  )
  \,.
\end{displaymath}
To express the correct action functional for the 7d Chern-Simons theory it is useful to define the shifted [[supergravity C-field]]

\begin{displaymath}
\hat a  := \hat G_4 - \frac{1}{2}\hat \mathbf{\lambda}
  \,,
\end{displaymath}
which the object whose [[equations of motion]] with respect to the 7d Chern-Simons theory are still $\hat a = 0$.

Then in terms of the original $\hat G_4$ the action functional for the [[holographic principle|holographic dual]] [[higher dimensional Chern-Simons theory|7d Chern-Simons theory]] reads

\begin{displaymath}
\exp(i S(\hat G_4))
  = 
  \exp(i \int_X
    \frac{1}{2}
    (
      \hat G_4 \cup \hat G_4  
      -
      (\frac{1}{2}\hat \mathbf{\lambda})^2
    )
  )
  \,.
\end{displaymath}
This is the action as it appears in (\hyperlink{Witten96}{Witten96, (3.6)}).

In terms of [[twisted differential c-structures]] we may summarize the outcome of this reasoning as follows:

\emph{The divisibility of the action functional requires a $2(G_4 - a)$-[[twisted Wu structure]] in $\mathbb{Z}/2$-cohomology. Its lift to integral cohomology is the $2(G_4 - a)$-[[twisted differential string structure]] known as the ``Witten quantization condition'' on the [[supergravity C-field]]}.

\hypertarget{RestrictionOfTheCField}{}\subsubsection*{{Restriction of the supergravity $C$-field}}\label{RestrictionOfTheCField}

We discuss the conditions on the restriction of the [[supergravity C-field]] on the ambient [[11-dimensional supergravity]] [[spacetime]] to the M5-brane.

This is similar to the analogous situation in [[type II string theory]]. The [[Freed-Witten anomaly cancellation]] condition demands that the restriction of the [[B-field]] $\hat H_3 \hat H^3(X)$ on spacetime $X$ to an [[orientation|oriented]] [[D-brane]] $Q \hookrightarrow X$ has to trivialize, up to [[torsion]], relative to the integral [[Stiefel-Whitney class]] $W_3 = \beta(w_2)$, where $\beta$ is the [[Bockstein homomorphism]] induced from the [[short exact sequence]] $\mathbb{Z} \stackrel{\cdot 2}{\to}
\mathbb{Z} \to \mathbb{Z}_2$:

\begin{displaymath}
H_3|_Q \simeq W_3
  \,,
\end{displaymath}
thus defining a [[twisted spin{\tt \symbol{94}}c-structure]] on the [[D-brane]].

The analog of this for the M5-brane is discussed in (\hyperlink{Witten2000}{Witten00, section 5}). There it is argued that there is a class

\begin{displaymath}
\theta \in H^3(Q, U(1))
\end{displaymath}
on the 5-brane such that under the [[Bockstein homomorphism]] $\beta'$ induced by the [[short exact sequence]] $\mathbb{Z} \to \mathbb{R} \to U(1)$ we have for the [[supergravity C-field]] $\hat G \in \hat H^4(X)$ the condition

\begin{displaymath}
G|_Q = \beta'(\theta)
  \,.
\end{displaymath}
By the \hyperlink{7dCSDual}{above} quantization condition, this may also be thought of as witnessing a [[twisted string structure]] on the 5-brane (\hyperlink{Sati10}{Sati}).

This condition reduces to the above one for the $B$-field under [[double dimensional reduction]] on the circle.

\hypertarget{m5brane_charge}{}\subsubsection*{{M5-brane charge}}\label{m5brane_charge}

See at \emph{[[M5-brane charge]]}

$\backslash$linebreak

\hypertarget{AnomalyCancellation}{}\subsubsection*{{Anomaly cancellation}}\label{AnomalyCancellation}

Consider an 11-dimensional [[spin structure|spin]]-[[manifold]] $X^{(11)}$ and a 2-parameter family of 6-dimensional [[submanifolds]] $Q_{M5} \hookrightarrow X^{(11)}$. When regarded as a family of [[worldvolumes]] of an [[M5-brane]], the family of [[normal bundles]] $N_X Q_{M5}$ of this inclusion carries a [[characteristic class]]

\begin{equation}
I^{M5}
  \;\coloneqq\;
  I^{M5}_{\psi} 
  + 
  I_{C}
  \;\in\;
  H^8(F \times Q_{M5},\mathbb{Z})
\label{IM5}\end{equation}
where

\begin{enumerate}%
\item the first summand is the class of the [[chiral anomaly]] of [[chiral fermions]] on $Q_{M5}$ (\hyperlink{Witten96}{Witten 96, (5.1)}),


\item the second term the class of the [[quantum anomaly]] of a [[self-dual higher gauge field]] (\hyperlink{Witten96}{Witten 96, (5.4)})



\end{enumerate}
Moreover, there is the restriction of the [[I8]]-term (see \href{I8#eq:TheTerm}{there}) to $Q_{M5}$, hence to the [[tangent bundle]] of $X^{11}$ to $Q_{M5}$ (the ``anomaly inflow'' from the [[bulk spacetime]] to the M5-brane)

\begin{equation}
I_8\vert_{M5}
  \;\coloneqq\;
   I_8
   \big(
     T_{Q_{M5}} X
   \big)
  \;\in\;
  H^8(F \times Q_{M5},\mathbb{Z})
  \,.
\label{I8AnomalyInflow}\end{equation}
The sum of these cohomology classes, evaluated on the [[fundamental class]] of $Q_{M5}$ is proportional to the [[second Pontryagin class]] of the [[normal bundle]]

\begin{equation}
I^{M5}
  \;+\;
  I_8\vert_{M5}
  \;=\;
  \tfrac{1}{24} p_2(N_{Q_{M5}})
\label{SumOfM5AndInflowAnomalyIsp2}\end{equation}
(\hyperlink{Witten96}{Witten 96 (5.7)})

This result used to be ``somewhat puzzling'' (\hyperlink{Witten96}{Witten 96, p. 35}) since consistency of the [[M5-brane]] in [[M-theory]] should require its total [[quantum anomaly]] to vanish. But $p_2(N_{Q_{M5}})$ does not in general vanish, and the right conditions to require under which it does vanish were ``not clear'' (\hyperlink{Witten96}{Witten 96, p. 37}).

(For more details on computations involved this and the following arguments, see also \hyperlink{BilalMetzger03}{Bilal-Metzger 03}).

A resolution was proposed in (\hyperlink{FreedHarveyMinasianMoore98}{Freed-Harvey-Minasian-Moore 98}), further clarified in (\hyperlink{Monnier13}{Monnier 13}), see also (\hyperlink{BahBonettiMinasianNardoni18}{Bah-Bonetti-Minasian-Nardoni 18}). There it is asserted that

\begin{enumerate}%
\item the correct bulk anomaly inflow is not just that from $I_8$ itself, but includes also a contribution from the class $G_4$ of the [[supergravity C-field]], as per \eqref{FiberIntegration} below (\hyperlink{Monnier13}{Monnier 13, around (3.11)});


\item for $G^{M5}_4$ the ``restriction'' of the class of the [[supergravity C-field]] to $Q_{M5}$, the term $I^{M5}_{C}$ in \eqref{IM5} should have a further summand $-\tfrac{1}{2}\big( G_4^{M5} \big)^2$ (\hyperlink{Monnier13}{Monnier 13, around (3.7)}, using \hyperlink{Monnier14b}{Monnier 14b, (2.13)})


\item for 11d spacetime a [[4-sphere]]-[[fiber bundle]],

\begin{displaymath}
\itexarray{
     S^4 &\longrightarrow& X^{(11)} 
     \\
     && \big\downarrow^{\mathrlap{\pi}}
     \\
     && X^{(6)}
  }
\end{displaymath}
as befits the [[near horizon geometry]] of a [[black brane|black]] [[M5-brane]], the [[supergravity C-field]] should be taken to be of the form (\hyperlink{Monnier13}{Monnier 13, (3.12)})

\begin{displaymath}
G_4 =\coloneqq \tfrac{1}{2}\chi + \pi^\ast(G^{M5}_4)
\end{displaymath}
with $\tfrac{1}{2}\chi$ the degree-4 [[Euler class]], whose integral over the 4-sphere fiber is unity (\href{Sullivan+model+of+a+spherical+fibration#SullivanModelForSphericalFibration}{this Prop.}), reflecting the presence of a single M5.



\end{enumerate}
By this proposal (also \hyperlink{BahBonettiMinasianNardoni18}{Bah-Bonetti-Minasian-Nardoni 18 (5)}, \hyperlink{BBMN19}{BBMN 19 (2.9) and appendix A.4, A.5}), the anomaly inflow from the bulk would not be just $I_8$, as in \eqref{I8AnomalyInflow} but would be all of the following [[fiber integration]]

\begin{equation}
\itexarray{
    \pi_\ast
    \Big(
      -
      \tfrac{1}{6}
      G_4  G_4  G_4
      +
      G_4  I_8
    \Big)
    & =
    - \tfrac{1}{24} p_2 + \tfrac{1}{2}(G^{M5}_4)^2 + I_8
  }
\label{FiberIntegration}\end{equation}
Here we used \href{Spin5#FiberIntegrationOfCupPowersOfChiOver4Sphere}{this Prop} to find that

\begin{displaymath}
\pi_\ast\big(
    \chi^3
  \big)
  \;=\;
  2 p_2
\end{displaymath}
which would cancel against the first term $\tfrac{1}{24} p_2$ in \eqref{FiberIntegration}. Hence with this proposal, the remaining M5-brane anomaly \eqref{SumOfM5AndInflowAnomalyIsp2} would be canceled.

$\backslash$linebreak

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Perry-Schwarz action]]


\item [[M5-brane instanton]]


\item [[half M5-brane]]


\item [[string theory]]


\item [[11-dimensional supergravity]], [[M-theory]]


\item [[supergravity Lie 6-algebra]], [[M-theory super Lie algebra]]



\end{itemize}
[[!include gauge theory from AdS-CFT -- table]]

[[!include table of branes]]

\hypertarget{References}{}\subsection*{{References}}\label{References}

\hypertarget{survey}{}\subsubsection*{{Survey}}\label{survey}

The history as of the 1990 is reviewed in

\begin{itemize}%
\item [[Mike Duff]], chapter II of \emph{[[The World in Eleven Dimensions]]: Supergravity, Supermembranes and M-theory}, IoP 1999 (\href{https://www.crcpress.com/The-World-in-Eleven-Dimensions-Supergravity-supermembranes-and-M-theory/Duff/9780750306720}{publisher})

\end{itemize}
Further reviews and general accounts include

\begin{itemize}%
\item [[Robbert Dijkgraaf]], \emph{The mathematics of fivebranes} (\href{http://arxiv.org/PS_cache/hep-th/pdf/9810/9810157v1.pdf}{pdf})


\item [[Hisham Sati]], \emph{[[Geometric and topological structures related to M-branes]]} (2010)



\end{itemize}
\hypertarget{ReferencesAsBlackBrane}{}\subsubsection*{{Black brane description}}\label{ReferencesAsBlackBrane}

\hypertarget{unwrapped}{}\paragraph*{{Unwrapped}}\label{unwrapped}

The M5 was first found as a [[black brane]] of [[11-dimensional supergravity]] (the [[black fivebrane]]) in

\begin{itemize}%
\item [[Rahmi Gueven]], \emph{Black $p$-brane solutions of $D = 11$ supergravity theory, Phys. Lett. B276 (1992) 49 and in [[Mike Duff]] (ed.),}[[The World in Eleven Dimensions]]\_ 135-141 (\href{http://inspirehep.net/record/338203?ln=en}{spire})

\end{itemize}
That this metric, as well as that of every black $p$ brane for \emph{odd} $p$, is completely non-singular was observed in

\begin{itemize}%
\item [[Gary Gibbons]], [[Gary Horowitz]], [[Paul Townsend]], p. 15 of \emph{Higher-dimensional resolution of dilatonic black hole singularities}, Class.Quant.Grav.12:297-318,1995 (\href{https://arxiv.org/abs/hep-th/9410073}{arXiv:hep-th/9410073})

\end{itemize}
Classification of more general M5-brane [[ADE-singularities]] is in

\begin{itemize}%
\item S. Ferrara, A. Kehagias, H. Partouche, A. Zaffaroni, \emph{Membranes and Fivebranes with Lower Supersymmetry and their AdS Supergravity Duals}, Phys.Lett. B431 (1998) 42-48 (\href{https://arxiv.org/abs/hep-th/9803109}{arxiv:hep-th/9803109})


\item Paul de Medeiros, [[José Figueroa-O'Farrill]], Section 8.3 of: \emph{Half-BPS M2-brane orbifolds}, Adv. Theor. Math. Phys. Volume 16, Number 5 (2012), 1349-1408. (\href{http://arxiv.org/abs/1007.4761}{arXiv:1007.4761}, \href{https://projecteuclid.org/euclid.atmp/1408561553}{Euclid})



\end{itemize}
also to some extent in

\begin{itemize}%
\item Changhyun Ahn, Kyungho Oh, Radu Tatar, \emph{Orbifolds $AdS_7 \times S^4$ and Six Dimensional $(0, 1)$ SCFT}, Phys. Lett. B442 (1998) 109-116 (\href{https://arxiv.org/abs/hep-th/9804093}{arXiv:hep-th/9804093})

\end{itemize}
but see p. 3 of

\begin{itemize}%
\item Michele Del Zotto, [[Jonathan Heckman]], [[Alessandro Tomasiello]], [[Cumrun Vafa]], \emph{6d Conformal Matter},  (\href{https://arxiv.org/abs/1407.6359}{arXiv:1407.6359})

\end{itemize}
Identification of the $\mathcal{N} = (2,0)$ black M5-brane sitting at the [[ADE singularity|A-type singularity]] of an $MO5$ $\mathbb{Z}/2$-[[orientifold]] locally of the form $\mathbb{R}^{5,1} \times ( \mathbb{R}^5 \sslash (\mathbb{Z}/2) )$ is due to

\begin{itemize}%
\item [[Edward Witten]], \emph{Five-branes And M-Theory On An Orbifold}, Nucl. Phys. B463:383-397, 1996 (\href{https://arxiv.org/abs/hep-th/9512219}{arXiv:hep-th/9512219})

\end{itemize}
Discussion in terms of [[E11]]-[[U-duality]] and [[current algebra]] is in

\begin{itemize}%
\item [[Hirotaka Sugawara]], \emph{Current Algebra Formulation of M-theory based on E11 Kac-Moody Algebra}, International Journal of Modern Physics A, Volume 32, Issue 05, 20 February 2017 (\href{https://arxiv.org/abs/1701.06894}{arXiv:1701.06894})


\item [[Shotaro Shiba]], [[Hirotaka Sugawara]], \emph{M2- and M5-branes in E11 Current Algebra Formulation of M-theory} (\href{https://arxiv.org/abs/1709.07169}{arXiv:1709.07169})



\end{itemize}
\hypertarget{wrapped_on_hyperblic_3manifolds}{}\paragraph*{{Wrapped on hyperblic 3-manifolds}}\label{wrapped_on_hyperblic_3manifolds}

Solutions to [[D=11 N=1 supergravity]] describing [[black brane|black]] [[M5-branes]] [[wrapped brane|wrapped]] on [[hyperbolic 3-manifolds]] (with application to the [[3d-3d correspondence]] and [[proof]] of the [[volume conjecture]]):

\begin{itemize}%
\item [[Jerome Gauntlett]], [[Nakwoo Kim]], [[Daniel Waldram]], Section 3.1 of: \emph{M-Fivebranes Wrapped on Supersymmetric Cycles}, Phys. Rev. D63 (2001) 126001 (\href{https://arxiv.org/abs/hep-th/0012195}{arXiv:hep-th/0012195})


\item Aristomenis Donos, [[Jerome Gauntlett]], [[Nakwoo Kim]], Oscar Varelam, \emph{Wrapped M5-branes, consistent truncations and AdS/CMT}, JHEP 1012:003, 2010 (\href{https://arxiv.org/abs/1009.3805}{arXiv:1009.3805})


\item Dongmin Gang, [[Nakwoo Kim]], Sangmin Lee, \emph{Holography of Wrapped M5-branes and Chern-Simons theory}, Physics Letters B Volume 733, 2 June 2014, Pages 316-319 (\href{https://arxiv.org/abs/1401.3595}{arXiv:1401.3595})


\item Dongmin Gang, [[Nakwoo Kim]], Sangmin Lee, \emph{Holography of 3d-3d correspondence at Large $N$}, JHEP04(2015) 091 (\href{https://arxiv.org/abs/1409.6206}{arXiv:1409.6206})


\item Dongmin Gang, [[Nakwoo Kim]], \emph{Large $N$ twisted partition functions in 3d-3d correspondence and Holography}, Phys. Rev. D 99, 021901 (2019) (\href{https://arxiv.org/abs/1808.02797}{arXiv:1808.02797})


\item Dongmin Gang, [[Nakwoo Kim]], Leopoldo A. Pando Zayas, \emph{Precision Microstate Counting for the Entropy of Wrapped M5-branes} (\href{https://arxiv.org/abs/1905.01559}{arXiv:1905.01559})



\end{itemize}
Discussion of the [[volume conjecture]] by combining the 3d/3d correspondence with [[AdS/CFT]] in these backgrounds:

\begin{itemize}%
\item \hyperlink{GangKimLee14b}{Gang-Kim-Lee 14b, Section 3.2}


\item \hyperlink{GangKim18}{Gang-Kim 18, around (21)}



\end{itemize}
Enhanced to a [[defect field theory]]:

\begin{itemize}%
\item Dongmin Gang, [[Nakwoo Kim]], Mauricio Romo, Masahito Yamazaki, \emph{Aspects of Defects in 3d-3d Correspondence}, J. High Energ. Phys. (2016) (\href{https://arxiv.org/abs/1510.05011}{arXiv:1510.05011})

\end{itemize}
\hypertarget{ReferencesSigmaModelDescription}{}\subsubsection*{{$\sigma$-Model description}}\label{ReferencesSigmaModelDescription}

The [[Green-Schwarz action functional]]-type [[sigma-model]] of the (single) M5-brane of was found in covariant form in

\begin{itemize}%
\item [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], \emph{Covariant Action for a D=11 Five-Brane with the Chiral Field}, Phys. Lett. B398 (1997) 41 (\href{https://arxiv.org/abs/hep-th/9701037}{arXiv:hep-th/9701037})


\item [[Igor Bandos]], [[Kurt Lechner]], [[Alexei Nurmagambetov]], [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], \emph{Covariant Action for the Super-Five-Brane of M-Theory}, Phys. Rev. Lett. 78 (1997) 4332-4334 (\href{http://arxiv.org/abs/hep-th/9701149}{arXiv:hep-th/9701149})



\end{itemize}
generally following

\begin{itemize}%
\item [[Paul Townsend]], section 3.3 of \emph{D-branes from M-branes}, Phys. Lett. B373 (1996) 68-75 (\href{https://arxiv.org/abs/hep-th/9512062}{arXiv:hep-th/9512062})

(which did not yet have the [[Hopf-Wess-Zumino term]])



\end{itemize}
and using the covariant mechanism for [[self-dual higher gauge fields]] from

\begin{itemize}%
\item [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], \emph{On Lorentz Invariant Actions for Chiral P-Forms}, Phys.Rev. D55 (1997) 6292-6298 (\href{https://arxiv.org/abs/hep-th/9611100}{arXiv:hep-th/9611100})

\end{itemize}
based on the non-covariant form of the [[self-dual higher gauge field|self-duality mechanism]] ([[Perry-Schwarz action]]) due to

\begin{itemize}%
\item [[Malcolm Perry]], [[John Schwarz]], \emph{Interacting Chiral Gauge Fields in Six Dimensions and Born-Infeld Theory}, Nucl. Phys. B489 (1997) 47-64 (\href{http://arxiv.org/abs/hep-th/9611065}{arXiv:hep-th/9611065})


\item [[John Schwarz]], \emph{Coupling a Self-Dual Tensor to Gravity in Six Dimensions}, Phys. Lett. B395:191-195, 1997 (\href{http://cds.cern.ch/record/317663}{cds:317663}, )


\item [[Mina Aganagic]], Jaemo Park, Costin Popescu, [[John Schwarz]], \emph{World-Volume Action of the M Theory Five-Brane}, Nucl.Phys. B496 (1997) 191-214 (\href{http://arxiv.org/abs/hep-th/9701166}{arXiv:hep-th/9701166})



\end{itemize}
Discussion of the equivalence of these superficially different action functionals is in

\begin{itemize}%
\item [[Igor Bandos]], [[Kurt Lechner]], [[Alexei Nurmagambetov]], [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], \emph{On the equivalence of different formulations of the M Theory five--brane}, Phys. Lett. B408 (1997) 135-141 (\href{http://arxiv.org/abs/hep-th/9703127}{arXiv:hep-th/9703127})

\end{itemize}
The [[equations of motion]] in [[super spacetime]] were derived in

\begin{itemize}%
\item [[Paul Howe]], [[Ergin Sezgin]], [[Peter West]], \emph{Covariant Field Equations of the M Theory Five-Brane}, Phys. Lett. B399 (1997) 49-59 (\href{https://arxiv.org/abs/hep-th/9702008}{arXiv:hep-th/9702008})

\end{itemize}
and using the [[superembedding approach]] in

\begin{itemize}%
\item [[Paul Howe]], [[Ergin Sezgin]], \emph{$D=11$, $p=5$}, Phys.Lett. B394 (1997) 62-66 (\href{https://arxiv.org/abs/hep-th/9611008}{arXiv:hep-th/9611008})

\end{itemize}
see

\begin{itemize}%
\item [[Dmitri Sorokin]], Section 5.2 of \emph{Superbranes and Superembeddings}, Phys.Rept.329:1-101, 2000 (\href{http://arxiv.org/abs/hep-th/9906142}{arXiv:hep-th/9906142})

\end{itemize}
A variant adapted to a 3+3-dimensional split in

\begin{itemize}%
\item Sheng-Lan Ko, [[Dmitri Sorokin]], Pichet Vanichchapongjaroen, \emph{The M5-brane action revisited}, JHEP11(2013)072 (\href{https://arxiv.org/abs/1308.2231}{arXiv:1308.2231})

\end{itemize}
The computation of the small fluctuations of this GS-type sigma-model around a solution embedding as the asymptotic boundary of the [[AdS-spacetime]] [[near-horizon geometry]] of a black 5-brane as \hyperlink{ReferencesAsBlackBrane}{above}, and the proof, to low order, that the result is the [[6d (2,0)-supersymmetric QFT]] appearing in [[AdS-CFT|AdS7-CFT6 duality]] is due to

\begin{itemize}%
\item P. Claus, [[Renata Kallosh]], [[Antoine Van Proeyen]], \emph{M 5-brane and superconformal $(0,2)$ tensor multiplet in 6 dimensions}, Nucl.Phys. B518 (1998) 117-150 (\href{http://arxiv.org/abs/hep-th/9711161}{arXiv:hep-th/9711161})

\end{itemize}
A review with emphasis on the coupling to the [[M2-brane]] is in

\begin{itemize}%
\item [[Ergin Sezgin]], P. Sundell, \emph{Aspects of the M5-Brane} (\href{http://arxiv.org/abs/hep-th/9902171}{arXiv:hep-th/9902171})

\end{itemize}
Further developments include

\begin{itemize}%
\item Sheng-Lan Ko, [[Dmitri Sorokin]], Pichet Vanichchapongjaroen, \emph{The M5-brane action revisited} (\href{http://arxiv.org/abs/1308.2231}{arXiv:1308.2231})

\end{itemize}
\hypertarget{ReferencesWorldvolumeTheory}{}\subsubsection*{{Worldvolume theory}}\label{ReferencesWorldvolumeTheory}

The original article suggesting the description of the [[self-dual higher gauge theory]] on the 5-brane [[holographic principle|holographically]] by a dual [[higher dimensional Chern-Simons theory]] is

\begin{itemize}%
\item [[Edward Witten]], \emph{Five-Brane Effective Action In M-Theory}, J.Geom.Phys.22:103-133,1997 (\href{http://arxiv.org/abs/hep-th/9610234}{arXiv:hep-th/9610234})

\end{itemize}
A precise mathematical formulation of the proposal made there is given in

\begin{itemize}%
\item [[Mike Hopkins]], [[Isadore Singer]], \emph{[[Quadratic Functions in Geometry, Topology, and M-Theory]]}

\end{itemize}
A discussion that embeds this argument into the larger context of [[AdS-CFT duality]] is in

\begin{itemize}%
\item [[Edward Witten]], \emph{AdS/CFT Correspondence And Topological Field Theory} JHEP 9812:012,1998 (\href{http://arxiv.org/abs/hep-th/9812012}{arXiv:hep-th/9812012})

\end{itemize}
Discussion of [[S-duality]] in 6d self-dual higher gauge theory via [[non-commutative geometry|non-commutative]]-deformation:

\begin{itemize}%
\item [[Varghese Mathai]], [[Hisham Sati]], \emph{Higher abelian gauge theory associated to gerbes on noncommutative deformed M5-branes and S-duality}, J. Geom. Phys. 92:240-251, 2015 (\href{https://arxiv.org/abs/1404.2257}{arXiv:1404.2257})

\end{itemize}
See also the references at \emph{[[6d (2,0)-supersymmetric QFT]]}.

The [[double dimensional reduction]] to the [[D4-brane]] [[D=5 super Yang-Mills theory]] and the relation to [[Khovanov homology]] is discussed in

\begin{itemize}%
\item [[Edward Witten]], \emph{Fivebranes and Knots} (\href{http://arxiv.org/abs/1101.3216}{arXiv:1101.3216})

\end{itemize}
with further comments in

\begin{itemize}%
\item Michele Nardelli, \emph{On some equations concerning Fivebranes and Knots, Wilson Loops in Chern-Simons Theory, cusp anomaly and integrability from String theory. Mathematical connections with some sectors of Number Theory} (2011) (\href{http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/nardelli2011b.pdf}{pdf})

\end{itemize}
A proposal for a construction as a [[higher gauge theory]] for [[differential string structure|string 2-connections]] is due to

\begin{itemize}%
\item [[Christian Saemann]], Lennart Schmidt, \emph{Towards an M5-Brane Model I: A 6d Superconformal Field Theory}, J. Math. Phys. 59 (2018) 043502 (\href{https://arxiv.org/abs/1712.06623}{arXiv:1712.06623})

\end{itemize}
based on

\begin{itemize}%
\item [[Christian Saemann]], Lennart Schmidt, \emph{The Non-Abelian Self-Dual String and the (2,0)-Theory} (\href{https://arxiv.org/abs/1705.02353}{arXiv:1705.02353})

\end{itemize}
\hypertarget{ReferencesHopfWessZuminoTerm}{}\subsubsection*{{Hopf-Wess-Zumino term}}\label{ReferencesHopfWessZuminoTerm}

The [[higher dimensional WZW model|higher WZW term]] of the M5-brane ([[Hopf-Wess-Zumino term]]) was first proposed in

\begin{itemize}%
\item [[Ofer Aharony]], p. 11 of \emph{String theory dualities from M theory}, Nucl. Phys. B476:470-483, 1996 (\href{https://arxiv.org/abs/hep-th/9604103}{arXiv:hep-th/9604103})

\end{itemize}
and had been settled by the time of

\begin{itemize}%
\item \hyperlink{BLNPST97}{BLNPST 97 (1)}.

\end{itemize}
The resemblence of the first summand of the term to the [[Whitehead integral formula]] for the [[Hopf invariant]] was noticed in

\begin{itemize}%
\item [[Kenneth Intriligator]], \emph{Anomaly Matching and a Hopf-Wess-Zumino Term in 6d, N=(2,0) Field Theories}, Nucl.Phys. B581 (2000) 257-273 (\href{https://arxiv.org/abs/hep-th/0001205}{arXiv:hep-th/0001205})

\end{itemize}
which hence introduced the terminology ``Hopf-Wess-Zumino term''. Followup to this terminology includes

\begin{itemize}%
\item Jussi Kalkkinen, [[Kellogg Stelle]], Section 3.2 of: \emph{Large Gauge Transformations in M-theory}, J. Geom. Phys. 48 (2003) 100-132 (\href{https://arxiv.org/abs/hep-th/0212081}{arXiv:hep-th/0212081})


\item Shan Hu, Dimitri Nanopoulos, \emph{Hopf-Wess-Zumino term in the effective action of the 6d, (2, 0) field theory revisted}, JHEP 1110:054, 2011 (\href{https://arxiv.org/abs/1110.0861}{arXiv:1110.0861})


\item [[Alex Arvanitakis]], Section 4.1 of \emph{Brane Wess-Zumino terms from AKSZ and exceptional generalised geometry as an $L_\infty$-algebroid} (\href{https://arxiv.org/abs/1804.07303}{arXiv:1804.07303})



\end{itemize}
More on the relation to the Hopf invariant in

\begin{itemize}%
\item [[Hisham Sati]], \emph{Framed M-branes, corners, and topological invariants}, J. Math. Phys. 59 (2018), 062304 (\href{http://arxiv.org/abs/1310.1060}{arXiv:1310.1060})

\end{itemize}
Discussion of the full 6d WZ term is in

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:Twisted Cohomotopy implies M5 WZ term level quantization]]}

\end{itemize}
\hypertarget{ReferencesAnomalyCancellation}{}\subsubsection*{{Anomaly cancellation}}\label{ReferencesAnomalyCancellation}

The original computation of the total M5-brane anomaly due to

\begin{itemize}%
\item \hyperlink{Witten96}{Witten 96}

\end{itemize}
left a remnant term of $\tfrac{1}{24} p_2$. It was argued in

\begin{itemize}%
\item [[Dan Freed]], [[Jeff Harvey]], [[Ruben Minasian]], [[Greg Moore]], \emph{Gravitational Anomaly Cancellation for M-Theory Fivebranes}, Adv.Theor.Math.Phys.2:601-618, 1998 (\href{https://arxiv.org/abs/hep-th/9803205}{arXiv:hep-th/9803205})


\item [[Jeff Harvey]], [[Ruben Minasian]], [[Greg Moore]], \emph{Non-abelian Tensor-multiplet Anomalies}, JHEP9809:004, 1998 (\href{https://arxiv.org/abs/hep-th/9808060}{arXiv:hep-th/9808060})


\item [[Adel Bilal]], Steffen Metzger, \emph{Anomaly cancellation in M-theory: a critical review}, Nucl.Phys. B675 (2003) 416-446 (\href{https://arxiv.org/abs/hep-th/0307152}{arXiv:hep-th/0307152})



\end{itemize}
that this term disappears (cancels) when properly taking into account the singularity of the [[supergravity C-field]] at the locus of the [[black brane|black]] [[M5-brane]].

A more transparent version of this argument was offered in

\begin{itemize}%
\item Samuel Monnier, \emph{Global gravitational anomaly cancellation for five-branes}, Advances in Theoretical and Mathematical Physics, Volume 19 (2015) 3 (\href{https://arxiv.org/abs/1310.2250}{arXiv:1310.2250})

\end{itemize}
based on a refined discussion of the quantum anomaly of the [[self-dual higher gauge field]] on the M5-brane in

\begin{itemize}%
\item Samuel Monnier, \emph{The anomaly line bundle of the self-dual field theory}, Comm. Math. Phys. 325 (2014) 41-72 (\href{http://arxiv.org/abs/1109.2904}{arXiv:1109.2904})


\item Samuel Monnier, \emph{The global gravitational anomaly of the self-dual field theory}, Comm. Math. Phys. 325 (2014) 73-104 (\href{http://arxiv.org/abs/1110.4639}{arXiv:1110.4639}, \href{http://www.physics.rutgers.edu/het/video/monnier11b.pdf}{pdf slides})



\end{itemize}
This formulation via an anomaly 12-form is (re-)derived also in

\begin{itemize}%
\item Ibrahima Bah, Federico Bonetti, [[Ruben Minasian]], Emily Nardoni, \emph{Class $\mathcal{S}$ Anomalies from M-theory Inflow}, Phys. Rev. D 99, 086020 (2019) (\href{https://arxiv.org/abs/1812.04016}{arXiv:1812.04016})


\item Ibrahima Bah, Federico Bonetti, [[Ruben Minasian]], Emily Nardoni, \emph{Anomaly Inflow for M5-branes on Punctured Riemann Surfaces} (\href{https://arxiv.org/abs/1904.07250}{arXiv:1904.07250})



\end{itemize}
\hypertarget{double_dimensional_reduction_to_d4brane}{}\subsubsection*{{Double dimensional reduction to D4-brane}}\label{double_dimensional_reduction_to_d4brane}

The relation of the M5-brane to the [[D4-brane]] and the [[D=5 super Yang-Mills theory]] in its [[worldvolume]] [[physical theory|theory]] by [[double dimensional reduction]]:

\begin{itemize}%
\item [[Eric Bergshoeff]], Mees de Roo, Tomas Ortin, \emph{The Eleven-dimensional Five-brane} (\href{http://astro.eldoc.ub.rug.nl/FILES/root/Preprints/1996/Eleven-dimensional/eleven-dimensional_five-brane.pdf}{pdf})


\item \hyperlink{APPS97a}{APPS 97a, Section 6}


\item [[Mina Aganagic]], Jaemo Park, Costin Popescu, [[John Schwarz]], Section 6 of \emph{Dual D-Brane Actions}, Nucl. Phys. B496 (1997) 215-230 (\href{https://arxiv.org/abs/hep-th/9702133}{arXiv:hep-th/9702133})


\item [[Neil Lambert]], Constantinos Papageorgakis, Maximilian Schmidt-Sommerfeld, \emph{M5-Branes, D4-Branes and Quantum 5D super-Yang-Mills}, JHEP 1101:083 (2011) (\href{http://arxiv.org/abs/1012.2882}{arXiv:1012.2882})


\item [[Chong-Sun Chu]], Sheng-Lan Ko, \emph{Non-abelian Action for Multiple Five-Branes with Self-Dual Tensors}, (\href{http://arxiv.org/abs/1203.4224}{arXiv:1203.4224}) JHEP05(2012)028


\item [[Neil Lambert]], Miles Owen, \emph{Charged Chiral Fermions from M5-Branes} (\href{https://arxiv.org/abs/1802.07766}{arXiv:1802.07766})



\end{itemize}
See also (\hyperlink{Witten11}{Witten 11}).

\hypertarget{open_m5branes}{}\subsubsection*{{Open M5-branes}}\label{open_m5branes}

Discussion of open M5-branes ending on [[M9-branes]] in a [[Yang monopole]] is in

\begin{itemize}%
\item [[Eric Bergshoeff]], [[Gary Gibbons]], [[Paul Townsend]], \emph{Open M5-branes}, Phys.Rev.Lett.97:231601 2006 (\href{http://arxiv.org/abs/hep-th/0607193}{arXiv:hep-th/0607193})

\end{itemize}
\hypertarget{nonabelian_2form_fields}{}\subsubsection*{{Nonabelian 2-form fields}}\label{nonabelian_2form_fields}

The fact that the worldvolume theory of the M5-brane should support fields that are [[self-dual higher gauge theory|self-dual]] [[connections on a 2-bundle]] ($\sim$ a [[gerbe]]) is discussed in

\begin{itemize}%
\item [[Edward Witten]], \emph{[[Conformal field theory in four and six dimensions]]}, in [[Ulrike Tillmann]], \emph{Topology, Geometry and Quantum Field Theory: Proceedings of the 2002 Oxford Symposium in Honour of the 60th Birthday of Graeme Segal}, London Mathematical Society Lecture Note Series (2004) (\href{http://arxiv.org/abs/0712.0157}{arXiv:0712.0157})

\end{itemize}
as well as sections 3 and 4 of

\begin{itemize}%
\item [[Edward Witten]], \emph{Geometric Langlands From Six Dimensions} (\href{http://arxiv.org/abs/0905.2720}{arXiv:0905.2720})

\end{itemize}
Proposals for how to implement this are for instance in

\begin{itemize}%
\item [[Chong-Sun Chu]], \emph{A Theory of Non-Abelian Tensor Gauge Field with Non-Abelian Gauge Symmetry $G \times G$} (\href{http://arxiv.org/abs/1108.5131}{arXiv:1108.5131})


\item [[Henning Samtleben]], [[Ergin Sezgin]], Robert Wimmer, \emph{(1,0) superconformal models in six dimensions} (\href{http://arxiv.org/abs/1108.4060}{arXiv:1108.4060})



\end{itemize}
A formal proposal is [[schreiber:7d Chern-Simons theory and the 5-brane|here]].

\hypertarget{more_on_the_holographic_description}{}\subsubsection*{{More on the holographic description}}\label{more_on_the_holographic_description}

\begin{itemize}%
\item [[Alexei Nurmagambetov]], I. Y. Park, \emph{On the M5 and the AdS7/CFT6 Correspondence} (\href{http://arxiv.org/abs/hep-th/0110192}{arXiv:hep-th/0110192})

\end{itemize}
\hypertarget{more_on_the_algebraic_topology}{}\subsubsection*{{More on the algebraic topology}}\label{more_on_the_algebraic_topology}

\begin{itemize}%
\item [[Edward Witten]], \emph{Duality relations among topological effects in string theory}, J. High Energy Phys. 0005 (2000) 031 (\href{http://arxiv.org/abs/hep-th/9912086}{arXiv:hep-th/9912086})


\item [[Hisham Sati]], \emph{Geometric and topological structures related to M-branes II: Twisted String and String{\tt \symbol{94}}c structures} (\href{http://arxiv.org/abs/1007.5419}{arXiv:1007.5419})


\item [[Hisham Sati]], \emph{Twisted topological structures related to M-branes II: Twisted $Wu$ and $Wu^c$ structures} (\href{http://arxiv.org/abs/1109.4461}{arXiv:1109.4461})



\end{itemize}
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